[vilhelm_s]

There is a lot of discussion at this Stack Exchange question, "Is Kline right that Cauchy believed that continuous functions must be differentiable?" (https://hsm.stackexchange.com/questions/3480/is-kline-right-...)

Mathematicians didn't state this as theorem or axiom, so it's hard to pin down exactly what they might have believed; the people answering the question talk about "except at isolated points" rather than "except at a finite number of points".